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Continuous Self-Maps of the Circle

Published online by Cambridge University Press:  20 November 2018

J. Schaer*
Affiliation:
Department of Mathematics and Statistics The University of Calgary 2500 University Drive N.W. Calgary, Alberta T2N 1N4
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Abstract

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Given a continuous map δ from the circle S to itself we want to find all self-maps σ: SS for which δ o σ. If the degree r of δ is not zero, the transformations σ form a subgroup of the cyclic group Cr. If r = 0, all such invertible transformations form a group isomorphic either to a cyclic group Cn or to a dihedral group Dn depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: Let Δ: ℝ → ℝ be continuous, nowhere constant, and limx→−∞ Δ(x) = −∞, limx→−∞ Δ(xx) = +∞; then the only continuous map Σ: R → R such that Δ o Σ = Δ is the identity Σ = id.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Bisztriczky, T. and Rival, I., Continuous, slope-preserving maps of simple closed curves, Canad. J. Math. Vol. XXXII, 5 (1980), 11021113.Google Scholar
2. Munkres, J.R., Topology, A First Course, Prentice Hall 1975.Google Scholar